'The Eyeball Theorem' printed from http://nrich.maths.org/

In the interactive diagram below, the blue circle $C_1$ has a centre $X$ and fixed radius $R$, while
the grey circle $C_2$ has a centre $Y$ and variable radius $r$. Two tangents to $C_2$ are drawn from
$X$, cutting $C_1$ at points $A$ and $B$ and just touching $C_2$ at $P$ and $Q$. Tangents to $C_1$ are
drawn from $Y$ cutting $C_2$ at points $C$ and $D$ and just touching $C_1$ at $R$ and $S$.

Click and drag the centres of the circles in the dynamic diagram below. As you change the distance
between the centres of the circles and the radii what do you notice about the chords $AB$ and $CD$

This dynamic image is drawn using Geogebra, free software and very easy to use. You can download your
own copy of Geogebra from
http://www.geogebra.org/cms/
together with a good help manual and
Quickstart
for beginners. You may be surprised at how easy it is to draw the dynamic diagram above for yourself.

Doing mathematics often involves observing and explaining properties of `invariance', that
is, what remains the same when the rest of the pattern changes according to certain rules that can
be described in mathematical terms. NRICH dynamic mathematics problems allow you to alter the diagrams
and change some properties, so that you can observe what remains invariant. This may lead you to a
conjecture that you can prove. Proving the result in the case of The Eyeball Theorem uses only similar
triangles.